The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwells equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwells equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.
| Identifer | oai:union.ndltd.org:ADTP/216769 |
| Date | January 2002 |
| Creators | Axelsson, Andreas, kax74@yahoo.se |
| Publisher | The Australian National University. School of Mathematical Sciences |
| Source Sets | Australiasian Digital Theses Program |
| Language | English |
| Detected Language | English |
| Rights | http://www.anu.edu.au/legal/copyrit.html), Copyright Andreas Axelsson |
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